Applied Math Seminar

Although random forests are commonly used for regression, our understanding
of the prediction error associated with random forest predictions of individual re-
sponses is relatively limited. We introduce a novel measure of this error and evaluate
its properties, comparing it with the out-of-bag mean of squared residuals estimator
that, to our knowledge, is the only measure of random forest prediction error that
has been introduced in the literature thus far. We show that our proposed estimator
provides an individualized estimate of the error associated with a particular random
forest prediction, while the out-of-bag mean of squared residuals estimator provides
a more general estimate of the random forest's prediction error as a whole. Through
simulations on benchmark and simulated datasets, we also demonstrate that both
estimators of prediction error may form the bases for valid random forest predic-
tion intervals. Empirically, these prediction intervals performed as well as quantile
regression forest prediction intervals.

Stabilizing unstable states is a challenging task for control engineers and, at the same time, it is an exciting problem for computational neuroscientists. The mathematics involve investigating the properties of delay, or functional, differential equations. The benchmark paradigm is the stabilization of the upright position of an inverted pendulum. This talk reviews 14 years of student research at the Claremont Colleges aimed at determining the nature of the control mechanisms for human expert stick balancing. For seated expert stick balancers the time delay is 0.23s, the shortest stick that can be balanced for 240s is 0.32m and there is a sensory dead zone of 1-3 degrees for the estimation of the vertical displacement angle in the sagittal plane. These observations motivate a switching-type, pendulum-cart model for balance control which utilizes an internal model to compensate for the time delay by predicting the sensory consequences of the stick's movements. Numerical simulations using the semi-discretization method suggest that the feedback gains are tuned near the edge of stability. For these choices of the feedback gains the cost function which takes into account the position of the fingertip and the corrective forces is minimized. Surprisingly this model also explains why for the most expert, the stick always falls!

This talk studies the dynamics of a thin viscous liquid film coating the inner or outer surface of a sphere in the presence of gravity, surface tension and Marangoni effects. We also allow the sphere to rotate around its vertical axis. The surface tension coefficient can be considered as a constant, or a function of temperature or surfactant concentration. An asymptotic model describing the evolution of the film thickness is derived based on the lubrication approximation.

When the surface tension coefficient is a constant, the model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. This thesis shows that the steady states are of three different types: uniformly positive film thickness, or the states with one or two dry zones on the sphere, depending on the relative strength of the centrifugal force to that of gravity. The transient dynamics in approaching those states are also described. This thesis also provides a constructive proof for the existence of non-negative weak solutions in a weighted Sobolev space.

When the surface tension coefficient is a non-constant function, an additional term representing the Marangoni effect is added to the equation. This thesis studies the cases when the surface tension gradient is due to an externally imposed temperature field or the presence of surfactant molecules. In the former case, we consider two different heating regimes with axial or radial thermal gradients and discuss the resulting dynamics. In the latter case, this thesis derives and studies a model for the coating flow inside the alveolar compartment of the lungs, taking into account the effect of pulmonary surfactant and its production and degradation. We derive a degenerate system of two coupled parabolic partial differential equations that describe the time evolution of the thickness of the coating film together with that of the surfactant concentration at the liquid-air interface. This thesis presents numerical simulations of the dynamics using parameter values consistent with experimental measurements.

where $c, b_0, m_0$ are given functions in appropriate $L^p$-spaces on a smooth bounded region $\Omega$ in $\mathbb{R}^N$, and $\lambda, \mu$ are real eigenparameters.

Here, $m_0$ is assumed to be strictly positive, $b_0$ may be sign-changing, and $\nu$ denotes the outward normal vector.

The weak formulation of this problem leads to an analysis of abstract eigencurve problems associated with triples $(a, b, m)$ of continuous symmetric bilinear forms on a real Hilbert space $V$.

In this talk I will decribe how the eigenpairs form {\em variational eigencurves} with nice properties. In particular, the curves satisfy some convexity properties that are easy to describe. For example, the first eigencurve is convex, i.e. any straight line intersects the curve at most twice, and the second eigencurve is a little less convex, i.e. any straight line interesects the curve at most four times. In general, any line intersects the $n$th eigencurve at most $2n$ times.

In the normal human heart, a specialized region of the heart
called the sinus node sets the rhythm of the entire heart. However, in
some circumstances the normal sinus rhythm is disrupted and abnormal
cardiac arrhythmias arise. This talk will give a quick introduction to
some rhythms that are particularly important in medicine and
interesting in mathematics. One rhythm, called atrial fibrillation, is
associated with an irregular rhythm. This rhythm is generally not
fatal, but leads to an increased risk for stroke. Other rhythms, such
as ventricular tachycardia and ventricular fibrillation are life
threatening or fatal. In this talk, directed towards a general
audience, I will give a brief introduction to reading
electrocardiograms and then describe some of the mathematical
approaches that are being used to diagnose, predict and control these
abnormal rhythms.